episciences.org_6059_1643074029
1643074029
episciences.org
raphael.tournoy+crossrefapi@ccsd.cnrs.fr
episciences.org
journal of Groups, complexity, cryptology
1869-6104
10.46298/journals/jgcc
https://gcc.episciences.org
03
02
2020
Volume 12, Issue 1
On the lattice of subgroups of a free group: complements and rank
Jordi
Delgado
Pedro V.
Silva
A $\vee$-complement of a subgroup $H \leqslant \mathbb{F}_n$ is a subgroup $K
\leqslant \mathbb{F}_n$ such that $H \vee K = \mathbb{F}_n$. If we also ask $K$
to have trivial intersection with $H$, then we say that $K$ is a
$\oplus$-complement of $H$. The minimum possible rank of a $\vee$-complement
(resp. $\oplus$-complement) of $H$ is called the $\vee$-corank (resp.
$\oplus$-corank) of $H$. We use Stallings automata to study these notions and
the relations between them. In particular, we characterize when complements
exist, compute the $\vee$-corank, and provide language-theoretical descriptions
of the sets of cyclic complements. Finally, we prove that the two notions of
corank coincide on subgroups that admit cyclic complements of both kinds.
03
02
2020
6059
arXiv:1905.12597
10.46298/jgcc.2020.12.1.6059
https://gcc.episciences.org/6059